# Engaging students: Solving logarithmic equations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Kelley Nguyen. Her topic: how to engage Algebra II or Precalculus students when solving logarithmic equations.

How could you as a teacher create an activity or project that involves your topic? (Flashcard/Match Game)

Because the rules behind logarithms can be mastered with practice, I believe an activity would help the students understand and master the concept. For an activity, I would create a matching game. It will include multiple cards that have logarithmic equations, as well as a match card with its solution or rewritten equation. For example:

The students would be in groups of 2-4 players. The deck of cards will be well-shuffled and laid out face down. Player 1 will turn over two cards and determine if they’re a match. If they’re a matching pair, the student will keep the two cards. If they are not, the player will turn the cards face down again and now it’s Player 2’s turn. If the Player 1 found a match, he/she will go again, following their first attempt. The other players should be observing and checking each other’s pairs to ensure that they are correct matches. They can also help each other in the process, i.e. coaching.

Another activity can also be done with logarithmic equation and solution cards. In this activity, there are 2-4 players in each group. Each player will receive five cards from the deck and the rest of the deck will be placed in the middle of the players in one stack and face down. The players are able to look at their cards and think of the solutions to them. Player 1 will turn the top card in the deck face up. If Player 1 has a matching card, he/she will take the card and start a stack of his/her matching pairs then draw a card from the deck. [Note: players will have five cards at all times.] If Player 1 does not have a match, each player will take a turn. If there is no match, Player 2 will then flip the second card and repeat the process. When all cards in the deck have been flipped over, turn the entire deck face down again and continue. The game will go on until all cards are match up. Whoever has the most matched pairs wins the game.

How can this topic be used in your students’ future courses in mathematics or science?

Logarithms are used frequently in chemistry when learning about acidity. In particular, the following equation describes a derivation of pH as the measure of acidity, as well as estimating the pH of a buffer solution and finding pH at equilibrium in acid-base reactions.

$pH = pK_\alpha + \log_{10} \left( \displaystyle \frac{ [A^-]}{[HA]} \right)$

There is also a time when logarithms are used in physics when working with the Beer-Lambert Law. The intensity of a light Io passing through a length of size l of a solution of concentration c is given as follows:

$\log (I_0 ll) = \epsilon c l$,

where $\epsilon$ is the absorption coefficient.

Another way logarithms are utilized is in science courses when students are to make predictions on the spread of disease in the world. This issue is greatly seen as the population grows dramatically, and using a logarithmic approach will allow the student to make a reasonable guesstimate.

Because students are introduced to logarithms at the end of Algebra II, they will work with them a lot in pre-calculus, as well as into calculus when dealing with trigonometric equations where there is a variable in the base and in the exponent.

How can technology be used to effectively engage students with this topic? (graphing calculator)

Although I think it’s easier to punch logarithmic equations into a calculator to get an answer, I still think that the students should conceptually learn why we come up with the answer. So, before allowing students to use calculators, make sure they know how we came up with the solutions. Once the students have mastered that concept, let them explore with their graphing calculators.

First, have the students put in the basic log function in Y1, then give them a log function with a transformation, whether it be a vertical shift, horizontal shift, or expansion, and store it into Y2. Ask the students to describe what they see.

Another way to utilize calculators with this topic is showing that the properties of logs are true, such as the addition rules of logarithmic equations being the log of the product of the arguments. You can also show the students how to change the base of a logarithmic equation on their calculators, since the standard log key is programmed at log10. That can be found when you click MATH and choice A in the first drop-down list.

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